Abstract

The model of body–vortex interactions, where the fluid flow is planar, ideal and unbounded, and the vortex is a point vortex, is studied. The body may have a constant circulation around it. The governing equations for the general case of a freely moving body of arbitrary shape and mass density and an arbitrary number of point vortices are presented. The case of a body and a single vortex is then investigated numerically in detail. In this paper, the body is a homogeneous, elliptical cylinder. For large body–vortex separations, the system behaves much like a vortex pair regardless of body shape. The case of a circle is integrable. As the body is made slightly elliptic, a chaotic region grows from an unstable relative equilibrium of the circle-vortex case. The case of a cylindrical body of any shape moving in fluid otherwise at rest is also integrable. A second transition to chaos arises from the limit between rocking and tumbling motion of the body known in this case. In both instances, the chaos may be detected both in the body motion and in the vortex motion. The effect of increasing body mass at a fixed body shape is to damp the chaos.

1. Introduction

There are many flow situations in nature and technology that involve a rigid body and a system of vortices, e.g. for steady flow, the well-known and ubiquitous vortex street wake named for von Kármán. The interaction of these vortices and the body producing them is a problem of great practical interest since it determines key aspects of the drag and lift experienced by the body. For some recent experiments showing the variety of wake structures possible behind a body in oscillatory motion, see Schnipper et al. (2009).

In general, real flows of this kind are (somewhat) three dimensional. Viscous effects are important both in producing the vortices and in eventually dissipating them. The vortices are usually most compact and clearly defined close to where they are produced, and then gradually lose definition. In this paper, we consider a time-honoured idealization of the two-dimensional version of body–vortex interaction that, in its essentials, goes back to von Kármán’s modelling of the vortex street wake: first, the fluid is assumed incompressible and inviscid. Second, we assume plane potential flow except for a finite number of point vortices with assigned and invariable circulations. These vortices are intended to model vortices that either were created by vortex shedding from the body, as described above, or were produced elsewhere in the fluid and now affect the body in question. A key feature of our modelling is that the body is free to move under the influence of the fluid with the embedded vortices, and through the linear and angular momentum imparted to it initially.

For greater realism, one could stipulate that the vortex patterns studied should be idealizations of the actual vortex wakes produced by real bodies. However, for the present, this link has been down-played, if not altogether severed: our vortices are placed somewhere around the body and their circulations are assigned ‘at will’. The purpose of this further idealization is to allow the study of the structure of the dynamical system consisting of a body and one or more vortices. We hope, of course, that in doing so we shall uncover regimes of motion and mechanisms that shed light on body–vortex interactions observed in real systems. This paper, in fact, is primarily concerned with the dynamics of a body and a single vortex moving under their mutual interaction.

The equations of motion of a body and one or more point vortices in ideal, planar fluid have been the subject of considerable interest in recent years. From an extensive literature, we cite the work of Borisov et al. (2007), which builds on a series of earlier papers (Ramodanov 2001, 2002; Borisov & Mamaev 2003; Borisov et al. 2003). The thrust of our work is most closely aligned with Borisov et al. (2007) where a single numerical exploration of chaos in body–vortex interactions (a Poincaré section—see below) is included, although the parameters and initial conditions used are not specified. These authors write: ‘It seems to be interesting to explore how the integrability and chaotic behaviour of the system vary with the value[s of the parameters].’ Two significant parts of such an exploration are contained in the present paper.

We also mention a more abstract derivation of the equations of motion of vortices interacting with a body in Shashikanth et al. (2002) and Shashikanth (2005). These papers elucidate the mathematical structure of the system of equations we are studying, but do not explore actual solutions.

Finally, we mention the papers of Kanso & Oskouei (2008) and Michelin & Llewellyn Smith (2009) and references therein, motivated by biological locomotion studies, where the equations of body–vortex interactions are again derived, but are then usually augmented in some way to address mechanisms such as vortex shedding. In summary, there is a considerable body of work on the derivation of the equations of motion for point vortices interacting with a moving body. There is considerably less work, analytical or numerical, on the nature of the resulting motion.

We are particularly interested in elucidating the transition to chaos in the body–vortex system under consideration. As we shall see, the chaotic motion of both the body and the vortex occurs already for just one vortex and one body. However, in this simplest case, integrable motion is also possible for a circular cylinder, for example. Even for a non-circular cylinder, the degree of chaos varies considerably depending on the shape and mass of the body and, for a given shape and mass, on the initial conditions. Such variations are potentially important for applications.

One would assume that as the number of vortices is increased, chaotic motion will prevail, although the degree of spatial organization of shed vortices into quasi-regular arrays (short segments of a vortex street) in many experiments (e.g. Andersen et al. 2005) and observations may suppress the appearance of chaos relative to what one sees when the positioning of the vortices is varied freely. From a theoretical point of view, this study is a hybrid between the studies of chaos in systems of a few point vortices (Aref 1983, 1985) and the chaos observed in the three-dimensional motion of a solid body in an ideal fluid otherwise at rest (Aref & Jones 1993).

2. Basic equations

As mentioned, the equations of motion for our idealized system have been considered by several authors. Since there are a number of subtle issues, we give a brief summary of the equations of motion and establish our notation. For more detailed derivations, we refer the reader to the literature cited.

(a) The velocity field

Consider an unbounded, ideal, planar fluid that contains N point vortices of constant circulations Γ1,…,ΓN. We shall think of the flow plane both as the ‘ordinary’ two-dimensional plane with Cartesian coordinates x and y and as the complex plane with z=x+iy. In the latter representation, the positions of the vortices at time t may be thought of as complex numbers z1(t),…,zN(t). We shall refer to these coordinates as the laboratory frame. Let there also be a movable rigid body, at time t represented by a contour , parametrized as z0(t)+f(eiχ)eiθ(t), 0≤χ<2π. Here, f(ζ) is a conformal mapping giving the physical body shape in terms of the parametrization of a unit circle in a secondary, complex ζ-plane. The point z0(t)=x0(t)+iy0(t) is the geometric centre of the body, and θ(t) gives the orientation of the body relative to the laboratory frame. The mapping f(ζ) exists for any body shape by Riemann’s mapping theorem. The ζ-plane will be called the mapped plane. We shall use both the z-plane, where the motion we wish to analyse takes place, and the ζ-plane, which is often analytically convenient.

We define also a body-fixed frame, a system of coordinates, , that follow the body in its motion, i.e. the origin of these coordinates is at z0(t) and the coordinate axes rotate with the body. The transformation of coordinates between the laboratory frame and the body-fixed frame may be summarized by
In addition to these two frames, we introduce a body-centred frame and an aligned frame. The body-centred frame is centred at z0 at any time but remains aligned with the laboratory frame. The aligned frame, on the other hand, has the same origin as the laboratory frame but it rotates so as to be aligned with the body-fixed frame at any instant. If denotes the position of a point with respect to the aligned frame, we have the coordinate transformation formulae
The aligned frame is particularly useful when expressing the flow induced by the motion of the body. The instantaneous translational velocity of the body in the laboratory frame is . It may be written in terms of its aligned frame components, and , as2.1a
It is and , as well as the angular velocity of the body around z0,2.1b
that enter as coefficients of the unit potentials (Lamb 1932) when writing the fluid velocity field produced by the motion of the body.

The combined velocity field owing to the moving body and the vortices is most simply given in the mapped plane and then transformed to the laboratory frame. The final result is2.2
The first three terms are the (complex conjugate of the) velocity field generated by the motion of the body. The functions and are, respectively, the unit complex potentials associated with translation at unit velocity along the first and second axes of the aligned frame. The function is the complex potential associated with rotation of the body around z0 with unit angular velocity. The primes on all three unit potentials, and on f, signify differentiation with respect to ζ. The three unit potentials are uniquely determined by the body shape, i.e. by the Riemann mapping function.

Since the fluid area outside the body is multiply connected, we have the freedom to assign a net circulation about the body. This net circulation around a loop enclosing the body, but not the vortices, we call Γ0 and the fourth term in the bracket in equation (2.2) gives its contribution to the velocity field.

For each vortex, we first have a contribution corresponding to an isolated vortex, Γα, in the mapped plane, . Second, in order to satisfy the impermeability condition, there is an image vortex of strength −Γα inside the mapped body contour as ascribed by the Milne–Thomson circle theorem. Finally, there is another image vortex of strength Γα at the body centre cancelling out the circulation around the body introduced with the first image vortex.

Note how the time only enters into u(z,t) in equation (2.2) via the time dependence of the 2N+6 dynamical variables and ω.

(b) The vortex motion

The vortices move owing to their mutually induced velocities and owing to the additional potential flow induced by the presence and through the motion of the body. The calculation of the motion of N point vortices in the presence of a stationary body is described by Lin (1943) who traces the earlier contributions by Kirchhoff and Routh. The body motion generates an additional advecting potential flow, as in equation (2.2), that is to be added, as given by Lin’s formalism. The velocity of a vortex may be calculated from the formula
where is given by equation (2.2). This equation expresses that the vortex ‘feels’ the induced velocity from all the other vortices, their images in the body, its own images in the body and from the potential flow owing to the motion of the body. The singular contribution from the vortex itself, however, is to be subtracted out. Straightforward calculation gives2.3
where the prime on the summation symbol means omission of the singular term (β=α). For simplicity of notation, we have omitted the explicit time dependence, and defined the total circulation around a loop enclosing the body and all vortices,

(c) The body motion

The simplest way to obtain the equations of motion for the body is to use conservation of linear and angular momentum for the full system of fluid plus body since no external forces or torques act. These integrals of motion arise as integrals over the entire plane,
2.4aand
2.4b
Here, ρ(z) is the mass density at position z and the usual vector product in the angular momentum has been written in complex notation.

Each expression may be split up into an integral over the fluid region (|ζ|>1) and one over the region occupied by the body (|ζ|≤1).

In the region occupied by the body, we simply have rigid body motion:
The mass distribution of the body may be non-uniform. The body mass, m, its centre of mass in the body-fixed frame, , and I0, the body moment of inertia with respect to z0, are given, respectively, by
These quantities are all independent of time and determined by the choice of body shape and density distribution.

The fluid density is assumed uniform and is designated ρf. The velocity field to be used in equations (2.4) in the fluid region is the field (2.2). Since the velocity field contains singularities and the domain of integration extends to infinity, caution must be exercised in performing the integrals in equations (2.4). The integrals are only conditionally convergent, which motivates the introduction of the linear impulse, P, and angular impulse L. These are, respectively, the linear and angular momenta from equations (2.4) with the singular terms removed. See Lamb (1932) for further elaboration of Kelvin’s theory of the impulse. The resulting conserved impulses may be written
2.5aand
2.5b
The quantities Ajk appearing here are the elements of the symmetric, constant added mass tensor, uniquely determined from f(ζ) via the complex unit potentials through the formulae
We exploit the freedom of orientation of the body-fixed coordinates to set A12=A21=0. This is always possible owing to the symmetry of the tensor. We have defined the aligned frame unit stream functions, , j=1,2,3. The unit stream functions giving the velocity field in the body-fixed frame are
which we recognize in the terms of equations (2.5) involving the vortices. Further, defining the virtual mass tensor,
we may now solve for the body linear and angular velocities in equations (2.5). Thus, these velocities are expressed in terms of the configuration variables, z1,…,zN,z0 and θ and the integrals of motion, P and L, which for given initial conditions we can calculate from equation (2.5). The result is2.6
Inserting this into equations (2.1), and combining with equations (2.3), we have a closed set of equations of motion for the 2N+3 body and vortex coordinates (x1,y1,…,xN,yN,x0,y0,θ). Exploiting the conservation of impulses, we have reduced the dimension of the original phase space from 2N+6 to 2N+3. Note that if the body size shrinks to zero, equations (2.5) reduce to the well-known impulse expressions for N+1 vortices of circulations Γ0,…,ΓN.

(d) Kinetic energy and Hamiltonian

In addition to the two components of the linear impulse and the angular impulse, there is one more conserved quantity, that is essentially the total kinetic energy of the fluid and body less the singular terms owing to the infinite extent of the domain and the singular nature of the point vortices. This energy may be written as2.7Borisov et al. (2007) show that by a suitable choice of independent variables, H becomes a Hamiltonian for both the body and vortex motion. There is an unfortunate typographical error in this work: the complex conjugation in the denominator of the last term is missing. This error appears to have propagated via Newton (2001) from Hasimoto et al. (1984), where it appears in the expression for the Hamiltonian for point vortex motion around a stationary body.

All numerical solutions in the following were computed using equations (2.1), (2.3) and (2.6). Since equations (2.5) express conservation of linear and angular impulse, these integrals of motion are exactly conserved by our numerics. The only remaining integral that needs to be monitored as we evolve the system in time is the energy (2.7). We have used the MATLAB ode45 solver with the absolute and relative error tolerances set to 10−9. This precision leads to a relative energy drift in the range 10−8 to 10−5 for all computations reported here.

3. Limits of integrability

To gain insight into the dynamical system under consideration we first survey some limiting cases where the issue of integrability or chaos is well understood.

Consider first the approximate view that arises by thinking of the body with a circulation around it as a vortex. This is particularly apt if the body and the vortex are far from one another, or equivalently, if the body is made very small. This problem of a body and a single vortex now approximates the two-vortex problem that is always integrable. The validity of this point of view is illustrated in §4a. We may also consider a body with two vortices in the flow field around it. Again, if the vortices stay at a great distance from the body compared with the body size, we approach the three-vortex problem, which is well known to be integrable (Aref 1983).

Even for the general problem of a body and a vortex, the space of parameters to be explored is quite large, and we shall only cover some of it in this paper. (In this case, we deviate from our general notation by using subscript ‘v’ for quantities pertaining to the vortex rather than subscript ‘1’, i.e. the vortex position is zv=xv+iyv and its circulation is Γv.) The freedom of choice for the units of length, time and mass may be exploited to get rid of some of the many parameters of the problem. Thus, we may choose the size of the body, R, as our unit of length. We may choose the circulation of the point vortex, Γv, to be 1, thereby setting our unit of time as R2/Γv. Finally, we may choose the density of the fluid, ρf, to be unity and thereby set our unit of mass. There remain two parameters in the problem, viz the body density, ρb, in units of the fluid density, and the circulation around the body, Γ0, now in units of the vortex strength. In addition to these two parameters, we need to specify initial conditions for the position, orientation, linear and angular velocity of the body, and the position of the vortex. We shall generally assume that the position of the body at t=0 is z0=0 with orientation θ=0, which is equivalent to choosing the laboratory reference frame. Since there are five initial conditions left, and ω(0), and the values of four constants of motion that may be determined from these, there is in general a one-parameter family of solutions having the same constants of motion. This we exploit in §4b to construct Poincaré sections in order to diagnose chaos in the system.

For a homogeneous circular cylinder, the rotational symmetry of the cylinder, i.e. the invariance of the problem to the orientation angle θ, provides an additional integral, the angular velocity of the body, ω. Thus, in addition to the linear and angular impulses and the Hamiltonian, ω is conserved, and the problem of one point vortex of any circulation and a rigid, homogeneous circular body of any mass is integrable. We may expect, therefore, that chaotic motion will arise when we break this symmetry by making the body eccentric. In §4c, we trace how chaos manifests itself as the invariance to θ is broken in this way.

Another way of destroying the θ-independence for a circular body is to choose a non-uniform mass distribution, thus moving its centre of mass away from its geometrical centre. The possible occurrence of chaos owing to such a displacement is certainly of interest from an applications point of view since many structures in nature and technology have . In terms of the virtual mass tensor, making a circle slightly elliptic leads to A11≠A22. For the non-uniform circle, on the other hand, the added mass tensor is unchanged, while the off-diagonal elements M13 and/or M23 of the virtual mass tensor become non-zero. We shall explore the variations of chaos in body–vortex dynamics owing to non-zero off-diagonal elements in the virtual mass tensor in a future publication. Here, we confine ourselves to homogeneous elliptical cylinders.

Yet another limit arises if the body is very heavy and so, in essence, unaffected by the presence of the fluid and the vortex. For a fixed body, the motion of a single vortex around it is integrable (Aref 1985). In the limit of infinite body mass, the translational and rotational velocities of the body are constants. The problem may be viewed in the body-fixed frame where the only difference from the fixed body case is the addition of a time-independent potential flow generated by the body motion. This problem is therefore also integrable. For a body that is free to move, interacting with a vortex, we might therefore expect increased regularity, although not necessarily integrability, as the body mass is increased. This limit is briefly investigated in §4d. For the case of the three-dimensional motion of a heavy body in an ideal fluid in potential flow, it was shown that increasing the mass of the body relative to the mass of the displaced fluid led to greater regularity (Aref & Jones 1993).

When addressing the issue of chaotic motion, we may focus on the motion of the vortex or the motion of the body. There is a priori the possibility that the vortex could move chaotically while the body moves in a regular fashion or vice versa. Even if both motions are formally chaotic, one can still speak of the motion of one constituent being ‘more regular’ than the other. For example, in the limit where the vortex has degenerated to a passively advected particle in the field of a freely moving two-dimensional body, we know that the body motion is integrable, as first shown by Kirchhoff, cf. Lamb (1932). The advection of a particle in the time-dependent flow field generated by the moving body is presumably non-integrable in general. Relatively little is known about this problem, save for integrable examples such as the study by Maxwell (1869) of the trajectories of advected particles in the field of a uniformly translating cylinder. The problem of the presence or absence of chaos in the advection of particles by a moving body will be addressed in a separate study. However, the possibility of chaotic motion already for a passive particle suggests that the motion of a single vortex interacting with a body is, in general, non-integrable.

There are also problems of one or more vortices, a body and an advected particle that may be explored from the point of view of integrability or chaos. We also postpone such investigations for the future.

In summary, the transition from integrability to chaos is expected to occur already for just one vortex interacting with a homogeneous moving body. It is this transition and its parametric dependence on body shape and mass that we have explored in the present paper.

4. Numerical explorations

We have confined ourselves to variations in the eccentricity of a body of elliptical shape and uniform mass density, which we denote ρb. Thus, ρb<1 corresponds to light or ‘bubble-like’ bodies (although our bodies do not deform), whereas ρb>1 corresponds to more conventional solid bodies embedded in the flow.

The body shape is given by the Riemann mapping,
For 0≤a≤1, this maps the unit circle in the ζ-plane to an ellipse in the -plane with foci at ±2aR and eccentricity 2a/(1+a2). The integrable case of a circular cylinder (of radius R) corresponds to a=0. For a=1, we obtain a flat plate connecting . The complex unit potentials associated with translation and rotation of an ellipse and the three diagonal elements of the added mass tensor are

(a) Vortex pair analogy for large body–vortex separations

To orient ourselves, four examples of the paths of vortex and body centre are shown in figure 1. We use the units of time, length and mass mentioned earlier, i.e. Γv=1, R=1 and ρf=1.

In figure 1a,b we have Γ0=−0.5 so that Γ≠0. In that case, the body–vortex system has a well-defined centre of vorticity, zcv=iP/ρfΓ, marked by the large black dot in the figure. In figure 1a, the body and vortex are started far from one another (compared with the body size), and so the velocity field induced by the body is dominated by the term proportional to Γ0 in equation (2.3); the terms owing to the body motion fall off more quickly with distance. To a first approximation, the body and vortex thus rotate around the centre of vorticity much as a vortex pair of strengths Γv and Γ0 would. In figure 1b, the body and vortex are started close to one another. Although the coupling of the vortex and body is now much stronger, making the orbits very irregular, there is still clearly an overall trend of moving around the centre of vorticity.

Figure 1c only differs from figure 1a in that Γ0=−1 such that Γ=0. The centre of vorticity, zcv, has moved off to infinity just as in the case of a vortex pair of opposite strengths. Indeed, when the body and vortex are distant, they move much like a pair of vortices of opposite strengths in the direction of P, which is along the x-axis for the chosen initial conditions. When the body–vortex pair with Γ=0 are started close together, as in figure 1d, their trajectories are apparently chaotic, but there is still a general trend of motion in the direction of P, i.e. in the positive x-direction. We shall return to the body and vortex trajectories later but first describe our major numerical diagnostic of chaos, the Poincaré section.

(b) Poincaré sections

The main case that we study in this paper is Γ=0. This corresponds to the circulations of the body and vortex that one would find if the vortex had been shed from the body, certainly a most important case from the point of view of applications. There are at least two Poincaré sections that present themselves naturally. One, which we refer to as the position Poincaré section or pps, pertains to the motion of the vortex. Examples appear in figures 4–6. The other, which we refer to as the velocity Poincaré section or vps, pertains to the motion of the body. In the pps, we plot the position of the vortex in the body-centred frame (xv−x0,yv−y0), every time the body returns to its original orientation, i.e. every time the angle θ returns to 0. In the vps, we plot the velocity components of the body in the laboratory frame, , every time θ=0. We have found it useful also to look at three-dimensional ‘sections’ by plotting in the pps and in the vps (figure 3). According to the formal Hamiltonian theory (Borisov et al. 2007), the case P=0 is Liouville integrable. In any event, for the initial conditions used for the sections constructed here, we have P≠0.

The initial conditions for the numerically calculated solution curves used to generate a Poincaré section are chosen such that they correspond to the same linear and angular impulses and energy but have different initial separations between the body and the vortex. If we insert the expressions (2.6) into the Hamiltonian (2.7), it becomes a function of ζv,z0,θ,P and L. Keeping all these quantities except ζv fixed in the resulting expression, we obtain an ‘energy landscape’, H(ζv), or equivalently , the contours of which correspond to initial vortex positions with the same energy and impulses. Examples of such contours for various values of a are shown in figure 2. The initial translational and angular velocities of the body required to keep the impulses fixed vary along such a contour. For a given initial vortex position, the required initial velocity may be calculated from equations (2.6). We stress that the contours in figure 2 are not vortex trajectories because, as the system evolves, the dynamical variables used to generate the contour change.

Energy landscapes for ρb=0.25, Γ=0 and (a) a=0, (b) 0.5 and (c) 0.8. Dashed lines are level curves through a hyperbolic stagnation point (all stagnation points are marked by open circles). Initial vortex positions used for Poincaré sections are marked by black dots.

The linear and angular impulses used in all Poincaré sections in this paper were calculated from an arbitrarily chosen ‘seed’ initial condition,4.1
These lead to a non-zero P (and thus to an overall motion of the body and vortex) in the negative y-direction. From figure 2, it is clear that the ‘energy landscape’ resulting from this choice of initial condition undergoes several bifurcations as a is varied between 0 and 1. We expect transitions between chaos and regularity to occur as the initial separation between the body and the vortex is varied. Our approach has been to calculate a large number of contours within the region seen in figure 2 and then pick our initial conditions along the contour that leads to the widest range of initial body–vortex separations (although with the restriction that the level curve does not come closer than 0.05 to the body—we felt that placing a point vortex very close to a rigid body would generate solutions that are artificial when compared with the trajectories of a vortex in a viscous fluid). The initial vortex positions found in this way are shown by black dots in figure 2.

It appears from figure 3 that all points from the different solutions lie on a two-dimensional surface in the three-dimensional space. This is a consequence of the choice Γ=0 that Borisov et al. (2007) show corresponds formally to the two components of the linear impulse being integrals in involution in the sense of Hamiltonian dynamics. This means that the eight-dimensional phase space spanned by may be reduced to a three-dimensional phase space instead of the reduction to a four-dimensional space that would generally be possible with four integrals of the motion. The surfaces in figure 3 are isosurfaces of θ in the two differently chosen reduced phase spaces. We have verified numerically that for Γ≠0, points of the three-dimensional Poincaré sections generally do not form two-dimensional surfaces.

Three-dimensional Poincaré sections with a=0.3 and ρb=0.25; (a) pps and (b) vps, for the same initial conditions. Different solution curves have been given different colours from a spectrum going from red for initial body angular velocity ω(0)>0, over black for ω(0)≈0 to blue for ω(0)<0.

Figure 3 also illustrates that the points of the Poincaré sections may be split into those for which ω≤0 and those for which ω>0. This separation conveniently allows us to plot projections of the three-dimensional pps onto the (xv−x0,yv−y0)-plane without encountering overlapping points from the (largely) regular and chaotic regions. We have generated arrays of pps for a matrix of (a,ρb) values: a=0(0.1)0.8 and ρb=0.1, 0.25(0.25)1.5, and 2.0. Space does not allow us to include all these figures. We shall, thus, be content to summarize our observations while illustrating some of the more striking aspects by select figures.

(c) Eccentricity variations

In figures 4–6, we show a series of pps with a varying between 0 and 0.8, and with ρb=0.25. Among the eight values of the body density explored, we find this to be the value that most vividly illustrates the transitions between chaos and regularity. (There is nothing special about the value, ρb=0.25. A different choice for the ‘seed’ initial condition in equation (4.1), determining the values of P and L used in the parameter scan, might have led to the a-scan at, say, ρb=1 exhibiting the most interesting transitions between chaos and regularity.)

pps for ρb=0.25 and (a,b) a=0, (c,d) 0.1 and (e,f) 0.2. Left (right) column shows points where ω<0 (ω>0). The large dots are the corresponding initial vortex positions. The body contour is shown by a thick line.

The first thing to note is that there is no sign of chaos in figure 4a,b. This is as it should be, since these figures correspond to the integrable case a=0. In fact, in these pps, the curves do trace out vortex trajectories in the body-centred frame, viz the trajectories of zv−z0. There are both regular curves that encircle the body and regular curves that stay on the right side of the body. We shall refer to curves in the pps that encircle the body as orbiting and to curves that stay on one side of the body as trapped. There is a hyperbolic point in figure 4a just to the right of the circle. This corresponds to an unstable relative equilibrium of the circle-vortex system. Already for a=0.1, figure 4c, where the deviation of the ellipse from a circle is hardly visible, we see signs of chaos. The chaos develops in a narrow band separating the orbiting curves from the trapped curves to the right of the body. As the eccentricity is increased, this chaotic region in the left column of figures 4–6, i.e. for ω<0, becomes larger and larger, eventually consuming most of the phase space shown.

Figure 7 shows the body and vortex trajectories for the two initial conditions in the chaotic band in figure 4e and for the regular motions on either side of it. In the trajectory shown at the top of figure 7, the vortex and body orbit one another in a regular manner. The next two chaotic solutions show vortex trajectories that alternate between orbiting and being trapped in an intermittent way. This ‘indecision’ appears to be the physical mechanism that leads to the scattering of the points in the pps and, thus, to the numerical diagnosis of chaos. Finally, the trajectory at the bottom of figure 7 shows the vortex and body moving along side by side with the vortex trapped on one side of the body.

The vortex (double curve) and body (black curve) trajectories for four of the solutions in figure 4e and 0≤t≤300. The selected orbits are (top) the innermost of the orbiting curves, (bottom) the outermost curve trapped to the right of the body and (middle) the two solutions in between in the chaotic band. The orbits are tilted 90° relative to the pps.

The presence of chaos in this region is presumably related to the unstable relative equilibrium that we saw in the integrable a=0 case (figure 4a). Clearly, the slightest disturbance of a vortex on any of the homoclinic orbits starting at this stagnation orbit can cause its path to change from orbiting to trapped or vice versa. The small additional velocity field generated by the rotating body when a is slightly above zero provides such a disturbance. As a is increased further, the intensity of the disturbance is increased. Orbits in an ever wider band around the relative equilibrium orbit are affected, apparently somewhat randomly, so that they alternate between orbiting and trapped motions. The ‘flipping’ between orbiting and trapped motion will depend on the size and direction of the velocity field from the body rotation at the moment where the vortex passes by the unstable equilibrium point. If one could obtain an analytical expression for the homoclinic orbit starting and ending at the hyperbolic point in the energy landscape for a=0, one could use the method of Melnikov to prove rigorously the occurrence of chaos as a becomes non-zero. To the best of our knowledge, such analytical solutions are not currently available.

In figure 4b, there is also a stagnation point separating trapped solution curves and orbiting curves. Yet, in the pps, we do not observe chaos at this transition until a is raised to 0.6 in figure 6b. This turns out to be related to our crude sampling of initial conditions along the energy contour. Indeed, as illustrated in figure 8, increasing the density of initial conditions near the transition in figure 5d reveals a narrow chaotic band also for a=0.4. One expects such chaotic bands to exist for all a>0 at all transitions between orbiting and trapped curves.

Zoom-in for a refined pps for a=0.4, ρb=0.25 and ω≥0. Initial conditions are concentrated near the transition between orbiting and trapped curves, i.e. between the two rightmost black dots in figure 5d.

A new feature appears in the pps as a is increased: trapped trajectories appear to the left of the body, e.g. at the periphery of figure 4c. These are a bit difficult to see as they form a very narrow band and actually also have points in figure 4d, where the body is rotating in the opposite direction. They are easier to see in the three-dimensional sections, e.g. for a=0.3 in figure 3b, where they are shown in black and cross the plane. They are separated from the orbiting curves with ω<0 (blue) and from the orbiting curves with ω>0 (red) by a chaotic band.

In figure 9, we plot body and vortex trajectories in and around this chaotic band for the first 400 time units, just a fraction of the time series used for the Poincaré sections. The first, third and fifth plots appear as regular in the Poincaré sections in figure 3, while the second and fourth are taken from the chaotic region. We do not see a transition in the body and vortex paths such as in figure 7 when crossing the chaotic band. Indeed, the chaos in the second and fourth set of trajectories only reveals itself as a slight jiggling in the mutual orbiting motion of the body and vortex. This transition is, in fact, not related to the relative motion of the body and vortex but rather to the body motion alone. In the absence of a vortex Kirchhoff’s solution, cf. Lamb (1932), shows that, depending on the initial conditions, the body will either rock or tumble, the equation for the body angular velocity being essentially the pendulum equation. Here again we have a limiting case where even a slight perturbation will decide which mode of motion the body chooses. The force and torque on the body from the vortex can provide the perturbation that is needed.

Vortex (double curve) and body (black) trajectories for the solutions in and around the central chaotic region in figure 3b for 0≤t≤400.

To make this point more clear, we show in figure 10 the time evolution of the orientation of the body, i.e. θ modulo 2π, for the trajectories shown in figure 9. The five traces of θ from top to bottom correspond to the five trajectories from left to right. In the top trace, we see monotonic clockwise body rotation. In the second trace from the top, we again have clockwise rotation but with a flip to counterclockwise rotation and later an increased amplitude of the rocking of the body. The middle trace solution shows regular clockwise rotation with a super-imposed small-amplitude periodic rocking. In the second to the last trace, the rocking is of higher amplitude and the sense of rotation shifts from clockwise on average to counterclockwise about a third into the time interval shown. The last trace shows regular counterclockwise body rotation with small-amplitude rocking during the entire time interval. These calculations illustrate a second mechanism for the evolution of chaotic motion in the system under consideration.

Time evolution of θ modulo 2π for the orbits in figure 9. Curves top to bottom correspond to trajectories left to right in figure 9.

(d) Mass variations

We close this section by illustrating the effect of body mass variations on the regularity of solutions. The Poincaré sections displayed in figure 11 were produced with a=0.8 and ρb=0.25,1.5,4 and 16. The vps is a little different from that considered previously in that the points are intersections with the arg surface instead of the θ=0 surface used earlier. As anticipated, the effect of the vortex on the body motion becomes less and less pronounced as we increase body mass. Thus, for ρb=0.25, all the solutions of the Poincaré section appear chaotic, while for ρb=16 all solutions appear regular. In the limit of infinite body mass, or equivalently, zero fluid density, the body will have constant linear and angular velocities. The labelling of the axes shows that the velocity variations for ρb=16 are smaller than for ρb=0.25 by an order of magnitude.

vps for a=0.8 and (a) ρb=0.25, (b) 1.5, (c) 4 and (d) 16. Dots are intersections of the solution curves with the arg surface.

5. Conclusions and outlook

We have shown by numerical experiments that the motion of a homogeneous elliptical body and a single point vortex in an unbounded, ideal fluid is in general chaotic. We have identified two mechanisms for the generation of chaos. The first arises from perturbing an unstable, relative equilibrium that exists in the limiting, integrable case of a circle and a vortex. The small vortex velocity generated by the body rotation, when the body is made slightly non-circular, serves as a time-dependent perturbation. The specific value at a given instant of this velocity field determines whether the vortex will orbit the body or be trapped on one side of it. The second mechanism arises from the limiting case between rocking and tumbling body motion when no vortex is present. In this case, the body motion is again integrable. Now the torque on the body from even a very weak vortex can provide a perturbation that determines whether the body will rock or tumble. This second mechanism may help in understanding the existence of a chaotic regime between the rocking and tumbling motion of a heavy falling plate.

As already discussed in §3, there are several further parameters to vary even in the problem of a body and a single vortex: the body may be asymmetric in shape, or have an asymmetric mass distribution (i.e. its centre of mass need not coincide with its geometrical centre). The body may have an imposed circulation about it. Furthermore, the number of vortices or the number of bodies may be increased. Finally, there are a host of problems involving the advection of passive particles in the flow fields induced by mutually interacting bodies and vortices. We intend to explore some of these problems armed with the insights gained by this study.

Acknowledgements

We thank Morten Brøns and other members of Fluid•DTU for comments and discussion. This work is supported by a Niels Bohr Visiting Professorship at the Technical University of Denmark sponsored by the Danish National Research Foundation.

1985Chaos in the dynamics of a few vortices-fundamentals and applicationsProc. of the Sixteenth Int. Congress of Theoretical and Applied Mechanics, Lyngby, Denmark1984 eds OlhoffN., NiordsonF. I.4368Amsterdam, The NetherlandsNorth-Holland Publishing Company